PHYSICAL REVIEW B 87, 245114 (2013)

Band structure tunability in MoS2 under interlayer compression: A DFT and GW study

C. Espejo*

Programa de Nanociencias y Nanotecnologıa, Centro de Investigacion y de Estudios Avanzados del I.P.N. (CINVESTAV),Libramiento Norponiente 2000, C.P. 76230 Queretaro, Mexico and

Departamento de Ciencias Basicas, Universidad de Bogota Jorge Tadeo Lozano, Carrera 4 22-61 Bogota, Distrito Capital, Colombia

T. Rangel†

Institute of Condensed Matter and Nanosciences (IMCN), NAPS, Universite Catholique de Louvain, Chemin des Etoiles 8,1348 Louvain-la-Neuve, Belgium

A. H. RomeroPhysics Department, West Virginia University, P.O. Box 6315, Morgantown, West Virginia 26506, USA,

Max Planck Institute for Microstructure physics, Weinberg 2, 06120, Germany, andUnidad Queretaro, CINVESTAV, Libramiento Norponiente 2000, CP 76230, Queretaro, Mexico

X. Gonze and G.-M. RignaneseInstitute of Condensed Matter and Nanosciences (IMCN), NAPS, Universite Catholique de Louvain, Chemin des Etoiles 8,

1348 Louvain-la-Neuve, Belgium and European Theoretical Spectroscopy Facility(Received 31 January 2013; published 17 June 2013)

The electronic band structures of MoS2 monolayer and 2H1 bulk polytype are studied within density-functionaltheory (DFT) and many-body perturbation theory (GW approximation). Interlayer van der Waals (vdW)interactions, responsible for bulk binding, are calculated with the postprocessing Wannier functions method.From both fat bands and Wannier functions analysis, it is shown that the transition from a direct band gap in themonolayer to an indirect band gap in bilayer or bulk systems is triggered by medium- to short-range electronicinteractions between adjacent layers, which arise at the equilibrium interlayer distance determined by the balancebetween vdW attraction and exchange repulsion. The semiconductor-to-semimetal (S-SM) transition is foundfrom both theoretical methods: around c = 10.7 A and c = 9.9 A for DFT and GW, respectively. A metallictransition is also observed for the interlayer distance c = 9.7 A. Dirac conelike band structures and linear bandsnear Fermi level are found for shorter c lattice parameter values. The VdW correction to total energy was usedto estimate the pressure at which S-SM transition takes place from a fitting to a model equation of state.

DOI: 10.1103/PhysRevB.87.245114 PACS number(s): 71.15.Mb, 71.10.−w, 71.30.+h, 71.20.−b

I. INTRODUCTION

New physical properties exhibited by matter down to thenano scale have raised enormous amounts of experimentaland theoretical work. The understanding of the behavior ofnanostructured materials and joined advances in synthesistechniques have allowed for the design of systems withdesirable properties.

Among the most promising new materials, let us highlighttwo-dimensional (2D) systems, such as graphene.1,2 Theirappeal is remarkable, not only for technology, but also forbasic science, since new phenomena have been discoveredand, in some cases, because 2D materials serve as almostideal systems where theoretical models can be tested againstempirical data.3 Electrons at the K point of the Brillouin zoneof graphene behave as massless Dirac fermions, demonstratinga condensed-matter system exhibiting quantum electrodynam-ical processes.4 High electron mobility, vanishing effectivemass, and anomalous quantum Hall effect have been observedin graphene, placing it at the top of promising materials fornano technological applications.5,6

Theoretical studies and experimental evidence on theremarkable properties of graphene have produced a renewedinterest in 2D crystals. An important source of these systemsare the transition-metal dichalcogenides (TMDs),7 and MoS2

is perhaps one the most distinguished representatives of this

family. Contrary to graphene, the finite band gap of MoS2

monolayer readily makes it suitable for electronic applicationssuch as the recently reported monolayer MoS2 transistor.8

It is well known that MoS2 undergoes a band-gap transitiongoing from direct gap in the isolated monolayer9–12 to indirectband gap in the case of bilayer or for bulk systems.13–15 Inprevious works,14,16 the band gap dependence upon both thenumber of layers and the interlayer distance was determined.On opposite sides of the scale we have the isolated monolayerand the equilibrium bulk. The evolution of the band gap wasmapped as a function of the number of layers, placing themat the experimental bulk interlayer distance. The result isthat the gap goes from direct to indirect while decreasing itsmagnitude from 1.8 to 0.86 eV.16 The latter values are notin total agreement with the experimental data15,16 given thetheoretical constrains which prevent DFT to predict band gapsaccurately. However, the global behavior is well described.Experimental realization of phototransistors which exploit thedependence of the gap with the number of layers has beenreported.17,18

Very recently, there have been theoretical reports demon-strating that MoS2 monolayers19,20 and bilayers undergo asemiconductor-to-semimetal (S-SM) transition under biaxialstrain21 and also compressive uniaxial strain in TMD bilayersystems.22 A similar behavior has also been predicted if an

245114-11098-0121/2013/87(24)/245114(8) ©2013 American Physical Society

ESPEJO, RANGEL, ROMERO, GONZE, AND RIGNANESE PHYSICAL REVIEW B 87, 245114 (2013)

electric field is applied perpendicular to the layers.23 Thesestudies are examples of the great band structure tunabilityfound on MoS2 few-layer systems.

Here we show that bulk also exhibit such S-SM transitionin the case of short interlayer distance, due to uniaxial strainperpendicular to the layers. We explore this additional degreeof freedom, which could be useful for band engineeringpurposes, from van der Waals-corrected DFT and MBPTcalculations. The inclusion of van der Waals interactionsis important in order to obtain an accurate description ofthe energetics involved in the uniaxial compression of thelayers.24 We found the appearance of an asymmetrical Diracconelike structure at the K point below the Fermi level atthe semimetal phase which is preserved for shorter interlayerdistances leading to a metallic transition.

The paper is organized as follows. The theoretical methodsand several calculation parameters are presented in the firstsection. Results and discussions are found next, devoting thefirst part to the structural characterization by using van derWaals-corrected DFT. Then the study of electronic structureas a function of c lattice parameter follows for two cases, onewhere the layer geometry is kept fixed at the monolayer valuesand a second one in which for each c value we perform in-planerelaxation of layers. MBPT results for the bulk are presentednext and concluding remarks are given in the last section.

II. THEORETICAL METHODS

A. vdW-corrected DFT and band structure

Van der Waals (vdW) interactions are crucial in thedetermination of the equilibrium configurations in layeredmaterials such as MoS2. These interactions are out of rangefor common used exchange-correlation (XC) functionals.25

Therefore, we have used our recent implementation in ABINIT26

of a method which makes it possible to calculate vdW energiesfrom maximally localized Wannier functions (MLWFs) of thesystem,27,28 also known as the vdW-WF method. This methodis based in the Anderson, Langreth and Lundqvist (ALL)functional29 which treats the long-range correlation energy,responsible for vdW interactions, as a postprocessing correc-tion to the ground-state energy. VdW energy is calculatedfrom a double spatial integral of the ground-state electronicdensity. In the vdW-WF method, the actual electronic densityis replaced with those ones coming from the Wannier functionsof the system which are, in turn, approximated by hydrogenlikefunctions, characterized by its spreads and centers. The latterapproximations allow for an important reduction of the compu-tational power needed to evaluate the double spatial integral ofthe ALL functional, resulting in an additional time comparableto a single step of the self-consistent field computation. Ingeneral, the spreads of MLWFs change only by a small amountwhen the ions are displaced by short distances. Therefore,for infinitesimal displacements of the interacting fragmentsthe spread change is negligible and they can be assumed tobe constant, giving raise to a computation method to evaluatethe vdW forces analytically, as it was clearly introducedin Ref. 28. Since the localization of Wannier functions isperformed through the interface to the WANNIER90 (Ref. 30)program, the inclusion of these forces into a geometry

optimization procedure is cumbersome and is not yet availablein ABINIT. Exchange and correlation energies are recom-mended to be treated with revPBE functional,31 avoidingspurious binding coming from exchange.27 However, we foundthat the vdW-WF correction to the results obtained with thePBE functional32 leads to equilibrium parameters which are inbetter agreement with experimental data for MoS2 than thoseobtained from the vdW-WF correction to revPBE calculations.VdW interactions calculation demands convergence of totalenergy as small as a few meV; therefore, a cutoff energy of50 Ha and a 8 × 8 × 2 Brillouin zone sampling were usedthroughout. Molybdenum semicore states 4s and 4p wereincluded for both structural and band-structure calculations.The corresponding atomic pseudopotentials were generatedwith the code APE.33 Starting from geometrical parametersfound in the literature for MoS2 monolayer,14 we performedstructural relaxation until the maximum force on each atomwas less than 10−5 Ha/bohr. Once the monolayer structure wasdetermined, we optimized the unit cell geometry of the solid bycomputing the interlayer binding energy as a function of c lat-tice parameter while keeping the layer geometry fixed at theirisolated configuration. In a second case, we perform in-planerelaxation for each value of c until stress tensor componentswere less than 0.001 Gpa, as opposed to previous calculationsfor bilayer systems where the layer geometry was kept fixed.22

The number of units cells along each lateral primitive vectorwas 20 and 3 unit cells normal to the layers in both cases, forwhich convergence of vdW corrections was reached.

B. MBPT band structure

MBPT is well established as an accurate method to predictband gaps in solids, based on the one-electron Green’s functionG and the screened Coulomb potential W . In this paper,we adopt the standard so-called G0W0 technique proposedin Ref. 34. The Kohn-Sham (KS) eigenvalues EKS aretaken as a zeroth-order perturbation to the quasiparticle (QP)eigenenergies EQP as

EQPi = EKS

i + ⟨ψKS

i

∣∣� − Vxc

∣∣ψKSi

⟩, (1)

where Vxc is the XC potential and the self-energy � is anon-Hermitian, frequency-dependent, and nonlocal operator.The non-Hermitian part of the self-energy, which gives riseto QP lifetimes, is neglected. The KS wave functions (ψKS)are considered to be close to the QP ones; hence, they are notmodified. Recent results of quasiparticle self-consistent GW

(QSGW) approach on bulk, monolayer, and bilayer MoS2 haveshown better agreement with absorption experiments35 thanG0W0 at a larger computational cost, though. The plasmonpole model (PPM) of Ref. 36 was used to simplify thefrequency dependency of �. Recently, it has been shown thatthe plasmon-pole approximation is suitable for systems withd electrons,37,38 as is the case with MoS2. This PPM achievesthe best agreement with more robust integration methods.39

In this work the relaxed DFT bulk structures (including vdWforces) were used. We found it crucial to include Mo semicorestates 4s and 4p to obtain physical QP energies. For the G0W0

calculations, DFT-LDA fhi98PP40 pseudopotentials were usedsince G0W0 on top of LDA has proven successful in predictingband gaps in weakly correlated materials.41

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BAND STRUCTURE TUNABILITY IN MoS2 UNDER . . . PHYSICAL REVIEW B 87, 245114 (2013)

In the G0W0 sums 300 bands were included. An energycutoff of 7.0 and 30 hartrees were used to generate thepolarizability and the exchange part of �, respectively. Thischoice of parameters achieves a convergence in the QPeigenvalues of 0.01 eV. Furthermore, the QP energies werecomputed for the same k mesh used in the ground-statecalculations. The MBPT band structures presented here wereinterpolated using MLWFs, as explained elsewhere.42

III. RESULTS AND DISCUSSION

First, we present our results for the structural optimizationand electronic structure calculations from vdW-corrected DFTand DFT-GGA methods respectively, as applied for the MoS2

monolayer and 2H1 bulk polytype. Then, the G0W0 methodis applied for the study of the S-SM transition predicted fromDFT.

A. Structural characterization from DFT + vdW − WF

MoS2 is a layered material composed of weakly bondedsheets of trigonal prismatic symmetry. Figure 1 shows theunit cell with six atoms and the Wannier centers used toevaluate the vdW correction to the energy. In Table I, thestructural parameters obtained for the monolayer and bulk forseveral values of c lattice parameters are displayed. Calculatedin-plane lattice parameters for the isolated monolayer arefound to be in good agreement with the correspondingexperimental bulk values, with a difference of only 0.53%.With small variations, this result is also obtained with severalXC functionals and from all-electron calculations11 as well.Therefore, it is customary that in previous studies on theelectronic structure of MoS2 monolayers, bulk, and or few-layer systems, the internal geometry of the layers was set equalto either the reported experimental bulk values14 or to relaxedmonolayer parameters.15 However, as has been reported, theelectronic structure is highly sensitive to both the internalstructural parameters of the layers and the interlayer distance,which may undergo large changes in the case of appliedstrain19–21 or applied electric fields.23 In the case of uniaxialstrain studied in this work, such large changes in geometry arealso important and determine the S-SM transition, as explainedbelow. In general, as long as the length of the c parameteris reduced, an increment of the cell parameters a and b is

FIG. 1. (Color online) (a) 2H1 MoS2 unit cell. S atoms (yellow)in one layer are right on top of Mo atoms (gray) of the second layer.The Wannier centers are represented as small black spheres. z is thevertical distance between S atoms (considered as the layer width).(b) Top view of the MoS2 crystal.

TABLE I. Calculated MoS2 monolayer and bulk structural pa-rameters and experimental values from Ref. 13. Evolution of in-planeeffective mass for holes and electrons is given in the two last columns.

a (A) c (A) z (A) m∗h/me m∗

e/me

1L 3.178 3.138 −1.490 1.280Bulk 3.253 10.7 2.999 −0.469 1.024

3.248 10.8 3.011 −0.507 1.0423.241 10.9 3.023 −0.677 1.0713.236 11.0 3.034 −0.716 1.1413.199 12.5 3.115 −1.744 1.601

Exp. 3.160 12.294 3.172

observed. On the other hand, the layer width z depends linearlywith c, except for c > 11.2 A, the starting point of saturationtowards the isolated monolayer value z = 3.138 A.

The energetics of interlayer bonding has two main ingredi-ents, namely, the attractive vdW interactions and the exchangerepulsion between layers. Figure 2 displays the interactionenergy per layer and per unit surface area from both PBE andthe vdW-corrected functional. Each curve has been calculatedtwice, either fixing the layer geometry to that of the isolatedmonolayer or allowing the layers to relax for each c value.The reference energy was defined as the total energy of theunit cell for c = 30 A upon in-plane relaxation. In both casesthe PBE curve has no minimum, indicating that from thislevel of theory the system would be unstable. On the otherhand, once we add the correlation energy calculated from thevdW-WF method (red triangles in Fig. 2), the obtained curvesshow a minimum at c ∼ 12.5 A. Theoretical results43 fromrandom phase approximation, vdW-DF, and PBE-D methodsare displayed for comparison. DFT + vdW-WF provides bothinterlayer distance and interplanar binding energy which arein close agreement to other approaches for vdW interactions inDFT. From the results, it is clear that the energy cost of layerrelaxation, the so-called stabilization energy, is an importantcomponent of the total energy which must be considered. InFig. 2 the stabilization energy is the energy difference between

10 12 14 16 18 20 22 24 26c (Å)

-40

-20

0

20

40

60

80

100

120

140

Inte

rpla

nar b

indi

ng e

nerg

y (m

eV)

PBE (no relax)vdW-WF (no relax)PBE (relax)vdW-WF (relax)FittingPBE-D vdW-DF1 vdW-DF2 RPA vdW energy

FIG. 2. (Color online) Interaction energy per layer and primitivesurface unit. Comparison with results from several DFT methods.Present calculations: PBE (no relaxation), PBE (relaxed), vdWcorrection energy and vdW-WF (also fit to the latter values). Otherresults from Ref. 43.

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the PBE (non-relaxed) and PBE (relaxed) curves for c = 30 A.Because of this energy, the vdW-WF-corrected interplanarbinding energy, for the fixed-layers case, becomes positiveat c ∼ 17 A. It is worth noticing that the vdW-corrected,relaxed-layers energy curve displays a small bump over thezero energy. This unphysical result could be attributed to asmall noncompensated repulsion between the layers comingfrom the long-range behavior of the PBE functional. From afitting to a quadratic function around the minimum we obtaina bulk modulus of 29.11 GPa, which compares with 39 GPaobtained from vdW-DF44 using a fixed-layers approach. Layerrelaxation produces a softening of both elastic constants andbulk modules with respect to the fixed-layers case.43 Cubicsplines are used to compute the derivative of unit cell totalenergy with respect to the volume V when assessing thepressure at which electronic transitions may occur (P =−∂U/∂V ).

B. Direct to indirect band-gap transition revisited

Several theoretical works have shown that the direct bandgap of MoS2 monolayer is replaced with an indirect bandgap in systems with more than one layer.11,15 This effectis proportional to both the number of layers consideredand its proximity.14 Experimental evidence which confirmstheoretical descriptions has also been obtained.9,13 A directband gap is then characteristic of isolated monolayers only.The presence of neighbor layers changes it for an indirect gapeven with just one additional layer.

Hence, two limiting cases are the MoS2 isolated monolayerand the bulk system. Calculated projected densities of states(PDOS) for these systems are displayed in Fig. 3. Eventhough the changes of PDOS are small when going from theisolated monolayer to the bulk case, band structure displaysthe gap transition. It is found that the main contribution to

0

0.2

0.4

0.6

0.8Mo-sS-s

012345

DO

S(e_ /e

V) Mo-p

S-p

-4 -2 0 2 4 6Energy (eV)

010203040 Mo-d

S-d

Mo-sS-s

Mo-pS-p

-4 -2 0 2 4 6Energy (eV)

Mo-dS-d

FIG. 3. (Color online) Projected densities of states for (left)monolayer and (right) bulk at equilibrium. Contributions of atomicstates of sulfur and molybdenum are in green and black, respectively.

valence-band maximum (VBM) and conduction-band mini-mum (CBM) of monolayer comes from Mo d states, as wellas for the bulk case, followed by the p states contribution. Onthe other hand, the contribution from sulfur states undergoes avery small change for the two systems.

Although it has been claimed that this effect is due to vdWinteractions,45 we are calculating it from a non-self-consistentmethod which neither affect nor modifies the electronic densityas it is originally obtained from the self-consistent GGAcalculation. Therefore, the change in the band gap would be theresult of electronic interactions different from vdW althoughthe proximity of layers is indeed caused by them.

C. Evolution of band structure as a functionof c lattice parameter

S-SM transition in MoS2 monolayer and bilayer systemsdue to biaxial strain has been reported previously.19–21 Inthese studies both tensile and compressive strains are appliedhomogeneously along each lattice vector parallel to the layers.S-SM transition is reported around 10% and 15% for tensileand compressive strain, respectively, in the monolayer.

In the case of bulk MoS2 there is an additional degreeof freedom for a similar transition, corresponding to theapplication of uniaxial compressive strain normal to the layers.As a consequence, the gradual reduction of interlayer distancefosters confinement of charge at the Mo planes while reducinglayer width. A similar behavior has been reported in the caseof applied electric fields normal to the layers.23

The resulting modification of electronic band structure ischaracterized by a progressive reduction of the indirect bandgap magnitude between the � point at the valence band and amidpoint between K and � in the conduction band. Figure 4shows the evolution of the band gap as a function of the c

lattice parameter. From these data it is estimated that the S-SMtransition will occur at c = 10.7 A, equivalent to a compressivestrain of about 14.4% (see Figs. 4 and 5). From the cubic splineperformed on the values of corrected interaction energy as a

9 9.5 10 10.5 11 11.5 12 12.5c Lattice parameter (Å)

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

2

2.4

Ban

d ga

p (e

V)

Relaxed layers Indirect Relaxed layers DirectFixed layers Indirect Fixed layers Direct GW indirectGW direct

FIG. 4. (Color online) Evolution of both direct and indirect bandgaps of 2H1 MoS2 with the c lattice parameter. Circles, DFT, relaxedlayers; triangles, DFT, fixed layers; squares, G0W0 on DFT relaxedlayers. Negative gap values indicate that the conduction band bottomis below the valence band top, showing the S-SM transition. Themetallic transition occurs when the direct gap goes to zero.

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BAND STRUCTURE TUNABILITY IN MoS2 UNDER . . . PHYSICAL REVIEW B 87, 245114 (2013)

-2

-1

0

1

2

E - E

F (e

V)

Γ M K Γ

-2

-1

0

1

2

E - E

F (e

V)

Γ M K Γ

c=12.5 Åc=12.5 Å

c=10.6 Å c=10.6 Å

FIG. 5. (Color online) Semiconductor-to-semimetal transition.(Left) Fixed-layers case, last valence band and first conduction bandin blue. (Right) Relaxing-layer case, last valence band and firstconduction band in red.

function of unit cell volume, our estimated pressure at whichthis electronic transition may occur is 29.11 GPa.

In case of further compression, the direct band gap on �

is reduced until a metallic state of null gap is reached. InFig. 4, a positive change in the slope for both the direct andthe indirect gap is observed once the system has come to thesemimetallic state. This trend is more marked for the case inwhich layers are free to relax than if its geometry is fixed. Whilethe slopes for both gaps are the same in the case of relaxinglayers, for the rigid-layers case the direct gap has a largerslope as compared to that of the indirect gap. Therefore, in thelatter case, the metallic transition is reached for a larger c ifcompared to relaxing layers. An additional interesting featureof the electronic structure evolution is the reduction of thein-plane effective masses and the appearance of straight bandsnear � and K at the Fermi level. For small-enough c, Diracconelike structures appear at three points of the first Brillouinzone in the rigid-layers case while a similar band structure isfound at � for relaxing layers. It should be noticed that in thelatter case a single-band Dirac cone is found at K , with itsvertex approximately at 1.3 meV below the Fermi level; seedashed circle of Fig. 6. This linear band is almost degeneratedwith a normal parabolic band. At the G0W0 level the latterstructure gives rise to a full Dirac cone for c = 8.84 A.

Even though we have simulated a bulk system, shortinterlayer distance fosters the confinement of charge atmonolayer planes, an expected effect on truly 2D crystals.This is only possible due to the weak chemical interactionsbetween adjacent layers preventing the formation of bonds. Aswas discussed before, there is a dependence of layer width onc, in a similar fashion as for the case of biaxial strain.19–21 Thisreduction is associated with a greater charge confinement alongthe Mo planes which, in turn, provokes electronic transitionsand eventually the appearance of conical bands.

-2

-1

0

1

E - E

F (e

V)

Γ M K Γ-3

-2

-1

0

1

E - E

F (e

V)

Γ M K Γ

c=10.2Å c=9.7Å

c=9.8Å c=9.5Å

FIG. 6. (Color online) Metallic transition. (Left) Fixed-layersgeometry. (Right) Relaxed layers. Valence bands are closer toconduction bands in the case of fixed-layers than in the relaxing-layerscase. There is a difference of 0.5 A in the transition-lattice parameterbetween both cases. Regions where Dirac conelike bands appear areenclosed by green circles.

D. G0W0 band structure

In Figs. 7 and 8, results from G0W0 calculations aredisplayed. The geometries obtained for structural relaxationsat the DFT level for each c value were used. One of themain characteristics of the G0W0-corrected band structuresis a general increment of gap values. The latter has as a

-2

-1

0

1

2

E-EV

BM (e

V)

Γ M K Γ

-2

-1

0

1

E - E

VB

M (e

V)

Γ M K Γ

c=12.5Å

c=10.6Å

c=9.7Å c=9.5Å

FIG. 7. (Color online) G0W0 band structures for several valuesof c lattice parameter. For each c value the obtained geometries at theDFT level upon in-plane relaxation were used. Energies referred tothe VBM.

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-2

-1

0

1

2

E-EF

(eV

)

Γ M K Γ

-2

-1

0

1

E - E

F (eV

)

Γ M K Γ

c=11.0Å

c=10.2Å

c=9.4Å

c=8.84Å

FIG. 8. (Color online) G0W0 band structures for several valuesof the c lattice parameter. For each c value the obtained geometries atthe DFT level upon in-plane relaxation were used. Energies referredto the Fermi level.

consequence that both the S-SM transition as well as theSM-M transition occur for shorter interlayer spacing, as canbe seen in Fig. 4. Comparison with DFT results shows that theconical structures remain only at the K point of the Brillouinzone. However, a set of linear bands and a discontinuity inthe second derivative of the dispersion relation appears in thesame symmetry point just after the semimetal transition, whilefor the DFT case, these characteristics arise once the systemreaches the metallic state. From MBPT calculation, the coneis placed at 0.5 eV below Fermi energy, while from DFT itis placed at K , 1.65 eV below the Fermi energy. Apart frombeing located below the Fermi level, the main difference withgraphene Dirac cones is that linear bands do not have the sameextension in reciprocal space around the K point for electronsand holes, as can be seen in Figs. 8 and 10 for c = 8.84 A.

(b)(a)

(d)(c)

FIG. 9. (Color online) Fermi surfaces for (a) bulk in equilibrium,c = 12.5 A; (b) semimetal state, c = 10.6 A; (c) metallic transitionc = 9.7 A; and (d) appearance of conical bands, c = 9.4 A.

Finally, in Fig. 9, we present the evolution of the Fermisurface, calculated at the DFT level for relaxed layers. TheFermi surface increases its complexity when going from thenormal semiconductor state up to the metallic transition. Forsemimetallic and metallic states, Fermi surface share a similarstructure with TlBiTe, which is also a semimetal with oneDirac cone [in that case located on � (Ref. 46)]. For energiesaround the Dirac point, there are closed Fermi surfaces with C6

symmetry, as observed in Fig. 9(d). Another reported materialexhibiting a single Dirac cone on � is the topological insulatorBi2Se3 (Ref. 47), which is a material composed of quintuplelayers weakly bonded by vdW interactions. Clearly, from ourcalculations, MoS2 exhibits a behavior resembling topologicalinsulators, induced by great proximity between layers. Bandstructures showed so far represent the dispersion relation alongBZ special lines; hence, a 3D representation of E(kx,ky) isdesirable. A close-up view of the Dirac pointlike in MoS2 forc = 8.84 A is displayed in Fig. 10.

From Fig. 10 it is evident that linear bands, formingconical shapes resembling Dirac cones of graphene, arenot only occurring along the special lines passing by K .For the case displayed and for the whole set of performedcalculations on relaxed layers, conical bands appear below

ky

ky

kx Δk

0.36

0.34

0.32

0.30

0.36

0.34

0.32

0.30

0.30 0.32 0.34 0.36

9.60

9.55

9.50

9.459.60

9.55

9.50

9.45

E (eV)

-0.04 -0.02 0 0.02 0.04

(a)

(b)

(c)

(d)

k(c)

k(d)

FIG. 10. (Color online) Illustration of the Dirac cone obtainedfrom G0W0 calculations for c = 8.84 A around K . Contour plot of theupper (a) and lower (b) parts of the cone: The isocurves are separatedby 0.005 eV. The tip of the cone (in black) is at kx = 0.334 andky = 0.347 with E ∼ 9.515 eV. The color scheme for the contours ofis reported in panel (c). In the upper part of the cone (a), the lowestenergies are in red (black); when increasing they turn to yellow (verylight gray), green (light gray), cyan (gray), and finally blue (black)for the highest values. In the lower part of the cone (b), it is just thereverse. Cuts through the cone are also reported in panels (c) and(d), following, respectively, the directions k(c) and k(d) as indicated inpanel (a). The cuts which show an intersection are obtained by takingthe axes which pass through the tip. In panel (c), the other curvesare obtained by translating the k(c) axis along the k(d) axis by stepsof 0.005. The cuts provide the energy (in eV) as a function of thedistance �k with respect to the tip along k(c). In panel (c), it is just thereverse. The warping effect is clearly demonstrated by the differencein the slopes of the cones in panels (c) and (d).

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BAND STRUCTURE TUNABILITY IN MoS2 UNDER . . . PHYSICAL REVIEW B 87, 245114 (2013)

EF . This prevents the linear bands and the Dirac-like pointfrom governing the overall electronic behavior of the system.The experimental identification of Dirac cones away from theFermi level is nevertheless possible, as done in Ref. 48.The only case with conical bands at EF was obtained atthe DFT level for c = 9.8 A using the monolayer geometry;see Fig. 6. Although this hypothetical system could exhibitsome properties associated with relativistic electrons, itsphysical realization is not feasible.

IV. CONCLUSIONS

We have computed the geometric and electronic propertiesof bulk MoS2 under pressure, using different methodologies.The vdW-WF method describes the bonding between MoS2

layers with an accuracy similar to other postprocessingapproaches like DFT-D. Its results compare well with thatfrom self-consistent vdW-DF, although the bulk modulus isunderestimated by ∼10 GPa. MoS2 band structure tunabilityunder uniaxial pressure was demonstrated from DFT andMBPT approaches. The evolution of both indirect and directband gaps was calculated as a function of the c latticeparameter. This procedure was realized for two cases: keepingfixed the layers geometry in its isolated configuration and

relaxing the layers at each c value. Semiconductor-to-semimetal transitions were found at c = 10.7 A for both casesat the DFT level, corresponding to a pressure of ∼30 GPa,while G0W0 calculations predict the S-SM transition to occurat c ∼ 9.9 A. Appearance of conical bands was demonstratedfrom both theoretical approaches and for the relaxing andfixed-layers cases. If the relaxed layers geometry is used, theDirac point is always placed below the Fermi energy.

ACKNOWLEDGMENTS

We would like to acknowledge technical support fromY. Pouillon, A. Jacques, and J.-M. Beuken. This work wassupported by the FRS-FNRS through FRFC Projects No.2.4.589.09.F and No. 2.4645.08, the Communaute francaisede Belgique, through the Action de Recherche Concertee07/12-003 “Nanosystemes hybrides metal-organiques,” theRegion Wallonne through WALL-ETSF Project No. 816849.A.H.R. recognizes the support of CONACYT Mexico underProject No. 152153, MICINN of Spain under Grant No.MAT2010-21270-C04-01/03, as well as the Marie-Curie Intra-European Fellowship. C.E. acknowledges financial supportfrom CONACYT Mexico and Universidad de Bogota JorgeTadeo Lozano.

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